How to compute the action probabilities with Thompson sampling in deep Q-learning?
So, how can I calculate $\mu(a)$ when using Thompson Sampling based on dropout?

The only way I could see this being calculated is if you iterate over all possible dropout combinations, or as an approximation sample say 100 or 1000 actions with different dropout, to get a rough distribution.
I don't think this is feasible for practical reasons (the agent will learn so much more slowly due to these calculations, you may as well abandon Thompson Sampling and use epsilon-greedy), and you will have to avoid using importance sampling if you also want to use action-selection techniques where there is no easy way to calculate a distribution.
Many forms of Q-learning do not use importance sampling. These typically just reset eligibility traces if the selected action is different from maximising action.